On CR Paneitz operators and CR pluriharmonic functions

DOI 10.1007/s00208-014-1151-2

Mathematische Annalen

On CR Paneitz operators and CR pluriharmonic functions

Chin-Yu Hsiao

Received: 29 May 2014 / Revised: 22 November 2014 / Published online: 2 December 2014

© Springer-Verlag Berlin Heidelberg 2014

Abstract Let (X,T^1,0X)be a compact orientable embeddable three dimensional strongly pseudoconvex CR manifold and let P be the associated CR Paneitz operator.

In this paper, we show that (I) P is self-adjoint and P hasL²closed range. LetNand be the associated partial inverse and the orthogonal projection onto Ker P respectively, thenNandenjoy some regularity properties. (II) LetPˆ andPˆ0be the space ofL² CR pluriharmonic functions and the space of real part of L² global CR functions respectively. LetSbe the associated Szegö projection and letτ,τ0be the orthogonal projections ontoPˆ andPˆ0respectively. Then, = S+S+F0,τ = S+S+F1, τ0 = S+S +F2, where F0,F1,F2 are smoothing operators on X. In particular, ,τ andτ0 are Fourier integral operators with complex phases andPˆ^⊥

Ker P, Pˆ₀^⊥Pˆ,Pˆ₀^⊥

Ker P are all finite dimensional subspaces of C^∞(X)(it is well- known thatPˆ0 ⊂ ˆP ⊂ Ker P). (III) Spec P is a discrete subset ofRand for every λ∈Spec P,λ=0,λis an eigenvalue of P and the associated eigenspaceH_λ(P)is a finite dimensional subspace ofC^∞(X).

Contents

1 Introduction and statement of the main results . . . . 904

1.1 The phaseϕ . . . . 909

2 Preliminaries . . . . 910

3 Microlocal analysis forb . . . . 912

C.-Y. Hsiao was partially supported by Taiwan Ministry of Science of Technology project 103-2115-M-001-001 and the Golden-Jade fellowship of Kenda Foundation.

C.-Y. Hsiao (

B

Institute of Mathematics, Academia Sinica, 6F, Astronomy-Mathematics Building, No. 1, Sec. 4, Roosevelt Road, Taipei 10617, Taiwan

e-mail: chsiao@math.sinica.edu.tw; chinyu.hsiao@gmail.com

4 Microlocal Hodge decomposition theorems for P and the proof of Theorem 1.2 . . . . 918 5 Spectral theory for P . . . . 926 References. . . . 929

1 Introduction and statement of the main results

Let(X,T^1,0X)be a compact orientable embeddable strongly pseudoconvex CR man- ifold of dimension three. Let P be the associated Paneitz operator and letPˆ be the space ofL²CR pluriharmonic functions. The operator P and the spacePˆ play impor- tant roles in CR embedding problems and CR conformal geometry (see [2–4,9]). The operator

P:Dom P⊂L²(X)→L²(X)

is a real, symmetric, fourth order non-hypoelliptic partial differential operator and Pˆ is an infinite dimensional subspace of L²(X). In CR embedding problems and CR conformal geometry, it is crucial to be able to answer the following fundamental analytic problems about P andPˆ(see [2–4,9]):

(I) Is P self-adjoint? Does P hasL²closed range? What is Spec P ?

(II) If we have Pu = f, where f is in some Sobolev space H^s(X),s ∈ Z, and u ⊥Ker P. Can we haveu∈ H^s(X), for somes∈Z?

(III) The kernel of P studied first by Hirachi [9] contains CR pluriharmonic func- tions(see also Lee [11]). The question asked by Hirachi is whether there is any- thing else in the kernel? In [4], they asked further that is the kernel of P a direct sum of a finite dimensional subspace with CR pluriharmonic functions?

(IV) Letbe the orthogonal projection onto Ker P and letτ be the orthogonal pro- jection ontoPˆ. Let(x,y)andτ(x,y)denote the distribution kernels ofand τ respectively. The Poperator introduced in Case and Yang [2] plays a critical role in CR conformal geometry. To understand the operator P, it is crucial to be able to know the exactly forms of(x,y)andτ(x,y).

The purpose of this work is to answer the above questions. On the other hand, in several complex variables, the study of the associated Szegö projectionS andτ are classical subjects. The operatorSis well-understood;Sis a Fourier integral operator with complex phase (see Boutet de Monvel-Sjöstrand [1,10]). But forτ, there are fewer results. In this paper, by using the Paneitz operator P, we could prove thatτ is also a complex Fourier integral operator andτ =S+S+F1, F1is a smoothing operator. It is quite interesting to see if the result hold in dimension≥5. We hope that the Paneitz operator P will be interesting for complex analysts and will be useful in several complex variables.

We now formulate the main results. We refer to Sect.2for some standard notations and terminology used here.

Let(X,T^1,0X)be a compact orientable 3-dimensional strongly pseudoconvex CR manifold, whereT^1,0X is a CR structure of X. We assume throughout that it is CR embeddable in some C^N, for some N ∈ N. Fix a contact formθ ∈ C^∞(X,T^∗X) compactable with the CR structure T^1,0X. Then,(X,T^1,0X, θ)is a 3-dimensional

pseudohermitian manifold. LetT ∈ C^∞(X,T X)be the real non-vanishing global vector field given by

dθ ,T ∧u =0, ∀u ∈T^1,0X⊕T^0,1X, θ ,T = −1.

Let · | · be the Hermitian inner product onCT Xgiven by Z1|Z2 = −1

2idθ , Z1∧Z2,Z1,Z2∈T^1,0X,

T^1,0X ⊥T^0,1X :=T^1,0X, T ⊥(T^1,0X⊕T^0,1X), T|T =1. The Hermitian metric · | · onCT Xinduces a Hermitian metric · | · onCT^∗X. Let T^∗0,1Xbe the bundle of(0,1)forms ofX. Takeθ∧dθbe the volume form onX, we then get natural inner products onC^∞(X)and^0,1(X):=C^∞(X,T^∗0,1X)induced byθ∧dθand · | · . We shall use(· | ·)to denote these inner products and use·to denote the corresponding norms. LetL²(X)andL²_(0,1)(X)denote the completions of C^∞(X)and^0,1(X)with respect to(· | ·)respectively. Let

b:=∂^∗_b∂b:C^∞(X)→C^∞(X)

be the Kohn Laplacian (see [10]), where∂b :C^∞(X) → ^0,1(X)is the tangential Cauchy–Riemann operator and ∂^∗b : ^0,1(X) → C^∞(X)is the formal adjoint of

∂b with respect to (· | ·). That is, (∂bf |g) = (f |∂^∗bg), for every f ∈ C^∞(X), g ∈^0,1(X).

LetPbe the set of all CR pluriharmonic functions onX. That is, P = {u ∈C^∞(X,R); ∀x0∈ X,there is a f ∈C^∞(X)

with∂bf =0 nearx0and Re f =unearx0}. (1.1) The Paneitz operator

P:C^∞(X)→C^∞(X)

can be characterized as follows (see section 4 in [2] and Lee [11]): P is a fourth order partial differential operator, real, symmetric,P ⊂Ker P and

Pf =bbf +L1◦L2f +L3f, ∀f ∈C^∞(X),

L1,L2,L3∈C^∞(X,T^1,0X⊕T^0,1X). (1.2) We extend P toL²space by

P:Dom P⊂L²(X)→L²(X), Dom P=

u ∈L²(X);Pu∈ L²(X)

. (1.3)

LetPˆ ⊂L²(X)be the completion ofPwith respect to(· | ·). Then, Pˆ ⊂Ker P.

Put

P0= {Re f ∈C^∞(X,R); f ∈C^∞(X)is a global CR function on X} and letPˆ0 ⊂ L²(X)be the completion ofP0with respect to(· | ·). It is clearly that Pˆ0⊂ ˆP ⊂Ker P. Let

τ :L²(X)→ ˆP,

τ0:L²(X)→ ˆP0, (1.4)

be the orthogonal projections.

We recall

Definition 1.1 SupposeQis a closed densely defined self-adjoint operator Q:DomQ⊂H →RanQ⊂H,

where H is a Hilbert space. Suppose thatQhas closed range. By the partial inverse ofQ, we mean the bounded operatorM : H→DomQsuch that

Q M+π =I onH, M Q+π =I on DomQ, whereπ :H →KerQis the orthogonal projection.

Let⊂ X be an open set. For any continuous operator A: C₀^∞()→ D(), throughout this paper, we write A≡0 (on) ifAis a smoothing operator on(see Sect.2).

The main purpose of this work is to prove the following Theorem 1.2 With the notations and assumptions above,

P:DomP⊂L²(X)→L²(X)

is self-adjoint andPhas L²closed range. Let N : L²(X)→ DomPbe the partial inverse and let:L²(X)→KerPbe the orthogonal projection. Then,

, τ, τ0:H^s(X)→ H^s(X)is continuous,∀s∈Z,

N : H^s(X)→ H^s+2(X)is continuous,∀s∈Z, (1.5)

≡τ on X, ≡τ0on X (1.6)

and the kernel(x,y)∈D(X×X)ofsatisfies (x,y)≡

_∞

0

e^iϕ(x,y)ta(x,y,t)dt+ _∞

0

e^−iϕ(x,y)ta(x,y,t)dt, (1.7) where

ϕ∈C^∞(X×X), Imϕ(x,y)≥0, dxϕ|x=y= −θ(x), ϕ(x,y)= −ϕ(y,x),

ϕ(x,y)=0if and only if x =y, (1.8) (see Theorem1.9and Theorem1.11for more properties of the phaseϕ), and

a(x,y,t)∈S_cl¹(X×X×]0,∞[), a(x,y,t)∼^∞

j=0

aj(x,y)t^1−j in S₁¹_,0(X×X×]0,∞[), aj(x,y)∈C^∞(X×X), j =0,1, . . . ,

a0(x,x)= 1

2π⁻ⁿ, ∀x∈ X. (1.9)

(See Sect. 2and Definition2.1for the precise meanings of the notation≡and the Hörmander symbol spaces S_cl¹(X×X×]0,∞[)and S₁¹_,0(X×X×]0,∞[).

Remark 1.3 With the notations and assumptions used in Theorem1.2, it is easy to see thatis real, that is=.

Remark 1.4 With the notations and assumptions used in Theorem 1.2, let S : L²(X) → Ker∂b be the Szegö projection. That is, S is the orthogonal projection onto Ker∂b=

u ∈L²(X);∂bu=0

with respect to(· | ·). In view of the proof of Theorem1.2(see Sect.4), we see that≡S+SonX.

We have the classical formulas _∞

0

e^{−t x}t^mdt= m!x^−m−1, ifm∈Z, m≥0,

(−1)^m

(−m−1)!x^−m−1

logx+c−₋m−1

1 1

j

, ifm∈Z, m<0.

(1.10) Herex=0, Rex≥0 andcis the Euler constant, i.e.c=limm→∞(m

1 1

j −logm).

Note that _∞

0

e^iϕ(x,y)t ∞

j=0

aj(x,y)t^1−jdt= lim

ε→0+

_∞

0

e^{−t(−i(ϕ(x,y)+iε))} ∞

j=0

aj(x,y)t^1−jdt.

(1.11) We have the following corollary of Theorem1.2

Corollary 1.5 With the notations and assumptions used in Theorem1.2, there exist F1,G1,∈C^∞(X×X)such that

(x,y)=F1(−iϕ(x,y))⁻²+G1log(−iϕ(x,y)) +F1(iϕ(x,y))⁻²+G1log(iϕ(x,y)).

Moreover, we have

F1=a0(x,y)+a1(x,y)(−iϕ(x,y))+ f1(x,y)(−iϕ(x,y))², G1≡

∞ 0

(−1)^k+1

k! a2+k(x,y)(−iϕ(x,y))^k, (1.12) where aj(x,y), j =0,1, . . ., are as in(1.9)and f1(x,y)∈C^∞(X×X).

Remark 1.6 It should be mentioned that Hirachi [8] derived the leading order asymp- totics for the Szegö kernel. The key feature of Hirachi’s work is the identification of the coefficient for the logarithm term of the Szegö kernel. From Hirachi’s result, we can determine the logarithm terms in Corollary1.5and it is possible to to derive the full asymptotics for(x,y)by Hirachi’s method.

Put

Pˆ^⊥:=

u∈ L²(X);(u| f)=0,∀f ∈ ˆP , Pˆ₀^⊥:=

v∈L²(X); (v|g)=0,∀g∈ ˆP0

.

From (1.6) and some standard argument in functional analysis (see Sect.4), we deduce Corollary 1.7 With the notations and assumptions above, we have

Pˆ^⊥

KerP⊂C^∞(X), Pˆ0^⊥

KerP⊂C^∞(X), Pˆ0^⊥

Pˆ ⊂C^∞(X)

andPˆ^⊥

KerP,Pˆ₀^⊥

KerP,Pˆ₀^⊥Pˆ are all finite dimensional.

We have the orthogonal decompositions Ker P= ˆP^⊥⊕

Pˆ^⊥ Ker P

,

Ker P= ˆP0^⊥⊕ Pˆ0^⊥

Ker P ,

Pˆ = ˆP0⊕ Pˆ0^⊥

Pˆ

. (1.13)

From Corollary1.7, we know thatPˆ^⊥

Ker P,Pˆ₀^⊥

Ker P,Pˆ₀^⊥Pˆ are all finite dimensional subsets ofC^∞(X).

Since P is self-adjoint, Spec P⊂R. In Sect.5, we establish spectral theory for P.

Theorem 1.8 With the notations and assumptions above, SpecPis a discrete subset inRand for everyλ∈SpecP,λ=0,λis an eigenvalue ofPand the eigenspace

H_λ(P):= {u ∈DomP;Pu=λu}

is a finite dimensional subspace of C^∞(X).

1.1 The phaseϕ

In this section, we collect some properties of the phase functionϕ. We refer the reader to [10] for the proofs.

The following result describes the phase functionϕin local coordinates.

Theorem 1.9 With the assumptions and notations used in Theorem1.2, for a given point x0∈ X , let{Z1}be an orthonormal frame of T^1,0X in a neighbourhood of x0, i.e.Lx0(Z1,Z1)=1. Take local coordinates x =(x1,x2,x3), z=x1+i x2, defined on some neighbourhood of x0such thatθ(x0)=d x3, x(x0)=0, and for some c∈C,

Z1= ∂

∂z −i z ∂

∂x3 −cx3 ∂

∂x3 +O(|x|²), where _∂^∂_z = ¹₂(_∂^∂_x1 −i_∂^∂_x

2). Set y = (y1,y2,y3), w = y1+i y2. Then, for ϕ in Theorem1.2, we have

Imϕ(x,y)≥c 2

j=1

xj−yj², c>0, (1.14)

in some neighbourhood of(0,0)and ϕ(x,y)= −x3+y3+i|z−w|²

+(i(zw−zw)+c(−zx3+wy3)

+c(−zx3+wy3))+(x3−y3)f(x,y)+O(|(x,y)|³), (1.15) where f is smooth and satisfies f(0,0)=0, f(x,y)= f(y,x).

Definition 1.10 With the assumptions and notations used in Theorem 1.2, let ϕ1(x,y), ϕ2(x,y) ∈ C^∞(X × X). We assume that ϕ1(x,y)and ϕ2(x,y) satisfy (1.8) and (1.14). We say thatϕ1(x,y)andϕ2(x,y)are equivalent on X if for any b1(x,y,t)∈ S_cl¹(X×X×]0,∞[)we can findb2(x,y,t)∈ S_cl¹(X×X×]0,∞[)such that

_∞

0

e^iϕ1(x,y)tb1(x,y,t)dt≡e^iϕ2(x,y)tb2(x,y,t)dt on X and vise versa.

We characterize the phaseϕ

Theorem 1.11 With the assumptions and notations used in Theorem 1.2, let ϕ1(x,y)∈C^∞(X×X). We assume thatϕ1(x,y)satisfies(1.8)and(1.14).ϕ1(x,y) andϕ(x,y)are equivalent on X in the sense of Definition1.10if and only if there is a function h ∈C^∞(X ×X)such thatϕ1(x,y)−h(x,y)ϕ(x,y)vanishes to infinite order at x=y, for every(x,x)∈X×X .

2 Preliminaries

We shall use the following notations: R is the set of real numbers, R₊ :=

{x∈R;x≥0},N = {1,2, . . .},N0 = N

{0}. An elementα = (α1, α2, α3)of N³₀will be called a multiindex, the size ofαis:|α| =α1+α2+α3. Form∈N, we writeα ∈ {1, . . . ,m}³ifαj ∈ {1, . . . ,m}, j = 1,2,3. We writex^α = x₁^α1x₂^α2x₃^α3, x=(x1,x2,x3),∂x^α =∂^αx1¹∂x^α2²∂^αx3³,∂xj = _∂^∂_xj,∂x^α =_∂^∂|α|_xα.

In this paper, we letT XandT^∗Xdenote the tangent bundle ofXand the cotangent bundle ofXrespectively. The complexified tangent bundle ofXand the complexified cotangent bundle ofXwill be denoted byCT XandCT^∗Xrespectively. We write ·,· to denote the pointwise duality betweenT XandT^∗X. We extend ·,· bilinearly to CT X×CT^∗X. LetEbe aC^∞vector bundle overX. The fiber ofEatx∈Xwill be denoted byEx. LetY ⊂Xbe an open set. From now on, the spaces of smooth sections of EoverY and distribution sections ofEoverY will be denoted byC^∞(Y,E)and D(Y,E)respectively. LetE(Y,E)be the subspace ofD(Y,E)whose elements have compact support inY. Form∈R, we letH^m(Y,E)denote the Sobolev space of order mof sections ofEoverY. Put

H_loc^m(Y,E)=

u∈D(Y,E);ϕu ∈H^m(Y,E),∀ϕ ∈C^∞₀ (Y) , H_comp^m (Y,E)=H_loc^m(Y,E)∩E(Y,E).

Let ⊂ X be an open set. If A : C₀^∞() → D()is continuous, we write KA(x,y)or A(x,y)to denote the distribution kernel of A. The following two state- ments are equivalent

(a) Ais continuous:E()→C^∞(), (b) KA∈C^∞(×).

If Asatisfies (a) or (b), we say that Ais smoothing. Let B :C₀^∞() → D()be a continuous operator. We write A ≡ B (on) if A−B is a smoothing operator.

We say thatAis properly supported if SuppKA⊂×is proper. That is, the two projections:tx :(x,y)∈ SuppKA →x ∈ ,ty :(x,y)∈SuppKA → y ∈are proper (i.e. the inverse images oftx andtyof all compact subsets ofare compact).

Let H(x,y) ∈ D(×). We write H to denote the unique continuous operator C₀^∞()→D()with distribution kernelH(x,y). In this work, we identifyHwith H(x,y).

We recall Hörmander symbol spaces

Definition 2.1 LetM ⊂R^Nbe an open set, 0≤ρ≤1, 0≤δ≤1,m∈R,N1∈N.

S_ρ,δ^m (M×R^N1)is the space of alla∈C^∞(M×R^N1)such that for all compactK M

and allα∈N^N₀,β∈N^N₀¹, there is a constantC>0 such that

∂x^α∂_θ^βa(z, θ)≤C(1+ |θ|)^{m−ρ|β|+δ|α|}, (x, θ)∈K×R^N1.

We say thatS_ρ,δ^m is the space of symbols of ordermtype(ρ, δ). Put S^−∞(M×R^N1):=

m∈R

S_ρ,δ^m (M×R^N1).

Letaj ∈ S_ρ,δ^mj(M ×R^N1), j = 0,1,2, . . .withmj → −∞, j → ∞. Then there existsa ∈S_ρ,δ^m0(M×R^N1)unique moduloS^−∞(M×R^N1), such thata−k−1

j=0aj ∈ S_ρ,δ^mk(M×R^N1)fork=0,1,2, . . ..

Ifaandaj have the properties above, we writea ∼_∞

j=0aj inS_ρ,δ^m0(M×R^N1).

LetS_cl^m(M×R^N1)be the space of all symbolsa(x, θ)∈S₁^m_,0(M×R^N1)with

a(x, θ)∼ ∞

j=0

am−j(x, θ)inS₁^m_,0(M×R^N1),

withak(x, θ) ∈ C^∞(M ×R^N1)positively homogeneous of degreek inθ, that is, ak(x, λθ)=λ^kak(x, θ),λ≥1,|θ| ≥1.

By using partition of unity, we extend the definitions above to the cases whenMis a smooth paracompact manifold and when we replaceM ×R^N1 byT^∗M.

Let⊂X be an open set. Leta(x, ξ)∈ S^k₁

2,¹2(T^∗). We can define A(x,y)= 1

(2π)³

e^ix−y,ξa(x, ξ)dξ as an oscillatory integral and we can show that

A:C₀^∞()→C^∞() is continuous and has unique continuous extension:

A:E()→D().

Definition 2.2 Letk ∈ R. A pseudodifferential operator of orderktype(₂¹,¹₂)is a continuous linear mapA:C₀^∞()→D()such that the distribution kernel ofAis

A(x,y)= 1 (2π)³

e^ix−y,ξa(x, ξ)dξ witha ∈ S^k1

2,¹₂(T^∗). We calla(x, ξ)the symbol of A. We shall write L^k1

2,¹₂()to denote the space of pseudodifferential operators of orderktype(¹,¹).

We recall the following classical result of Calderon–Vaillancourt (see chapter XVIII of Hörmander [7]).

Proposition 2.3 If A∈L^k₁

2,¹2

(). Then,

A:H_comp^s ()→ H_loc^s−k()

is continuous, for all s∈R. Moreover, if A is properly supported, then A:H_loc^s ()→ H_loc^s−k()

is continuous, for all s∈R.

3 Microlocal analysis forb

We will reduce the analysis of the Paneitz operator to the analysis of Kohn Laplacian.

We extend∂btoL²space by∂b:Dom∂b⊂L²(X)→L²_(0,1)(X), where Dom∂b:=

{u∈L²(X); ∂bu ∈L²_(0,1)(X)}. Let

∂^∗b:Dom∂^∗b⊂L²_(0,1)(X)→L²(X)

be theL²adjoint of∂b. The Gaffney extension of Kohn Laplacian is given by b =∂^∗_b∂b:Domb⊂L²(X)→ L²(X),

Domb: =

u∈ L²(X);u∈Dom∂b, ∂bu ∈Dom∂^∗b

. (3.1)

It is well-known thatbis a positive self-adjoint operator. Moreover, the characteristic manifold ofbis given by

=

(x, ξ)∈T^∗X; ξ =λθ(x), λ=0

. (3.2)

Since X is embeddable,bhas L²closed range. LetG : L²(X)→ Dombbe the partial inverse and letS : L²(X)→Kerbbe the orthogonal projection (Szegö projection). Then,

bG+S=I onL²(X),

Gb+S=I on Domb. (3.3)

In [10], we proved thatG∈ L⁻₁¹

2,¹2

(X),S ∈L⁰₁

2,¹2

(X)and we got explicit formulas of the kernelsG(x,y)andS(x,y).

We introduce some notations. Let⊂Xbe an open set with real local coordinates x=(x1,x2,x3). Let f,g ∈C^∞(). We write f gif for every compact setK ⊂ there is a constantcK >0 such that f ≤cKgandg≤cKf onK. We need

Definition 3.1 a(t,x, η) ∈ C^∞(R₊×T^∗) is quasi-homogeneous of degree j if a(t,x, λη)=λ^ja(λt,x, η)for allλ >0.

We introduce some symbol classes

Definition 3.2 Letμ > 0. We say thata(t,x, η)∈ S_μ^m(R₊×T^∗)ifa(t,x, η) ∈ C^∞(R+×T^∗)and there is aa(x, η)∈S^m_1,0(T^∗)such that for all indicesα, β ∈N³₀, γ ∈N0, every compact setK , there exists a constantc_α,β,γ >0 independent of tsuch that for allt ∈R₊,

∂t^γ∂_x^α∂_η^β(a(t,x, η)−a(x, η))≤c_α,β,γe^−tμ|η|(1+ |η|)^m+γ−|β|, x∈ K,|η| ≥1.

The following is well-known (see [10])

Theorem 3.3 With the assumptions and notations above, G ∈ L⁻₁¹

2,¹2(X), S ∈ L⁰1

2,¹₂(X), S(x,y) ≡

e^iϕ(x,y)ta(x,y,t)dt , whereϕ(x,y) ∈ C^∞(X ×X)is as in (1.8)and

a(x,y,t)∈S_cl¹(X×X×]0,∞[), a(x,y,t)∼

∞ j=0

aj(x,y)t^1−j in S₁¹_,0(X×X×]0,∞[), aj(x,y)∈C^∞(X×X), j =0,1, . . . ,

a0(x,x)= 1

2π⁻ⁿ, ∀x∈ X,

and on every open local coordinate patch ⊂ X with real local coordinates x=(x1,x2,x3), we have

G(x,y)≡ ^∞

0

e^{i(ψ(t,x,η)−y,η)}−t

iψt(t,x, η)a(t,x, η)+∂a

∂t(t,x, η)

dt dη, (3.4) where a(t,x, η)∈ S_μ⁰(R₊×T^∗),ψ(t,x, η) ∈S_μ¹(R₊×T^∗)for someμ >0, ψ(t,x, η)is quasi-homogeneous of degree1,ψt(t,x, η) = ^∂ψ_∂t (t,x, η),ψ(0,x, η)

= x, η, Imψ≥0with equality precisely on({0} ×T^∗\0)

(R+×), ψ(t,x, η)= x, ηon, dx,η(ψ− x, η)=0on, and

Imψ(t,x, η)

|η| t|η|

1+t|η| dist

x, η

|η|

,

2

, t ≥0, |η| ≥1. (3.5) (See Theorem3.4below for the meaning of the integral(3.4).)

Proof We only sketch the proof. For all the details, we refer the reader to Part I in [10]. We use the heat equation method. We work with some real local coordinates x=(x1,x2,x3)defined on. We consider the problem

(∂t+b)u(t,x)=0 inR₊×,

u(0,x)=v(x). (3.6)

We look for an approximate solution of (3.6) of the formu(t,x)= A(t)v(x), A(t)v(x)= 1

(2π)³

e^{i(ψ(t,x,η)−y,η)}α(t,x, η)v(y)d ydη (3.7) where formally

α(t,x, η)∼^∞

j=0

αj(t,x, η),

withαj(t,x, η)quasi-homogeneous of degree−j. The full symbol ofbequals₂

j=0pj(x, ξ), where pj(x, ξ)is positively homo- geneous of order 2− jin the sense that

pj(x, λη)=λ^2−jpj(x, η), |η| ≥1, λ≥1.

We apply∂t+bformally inside the integral in (3.7) and then introduce the asymptotic expansion ofb(αe^iψ). Set(∂t+b)(αe^iψ)∼0 and regroup the terms according to the degree of quasi-homogeneity. The phaseψ(t,x, η)should solve

∂ψ∂t −i p0(x, ψ_x)=O(|Imψ|^N), ∀N≥0,

ψ|t=0= x, η, (3.8)

where ψx = (^∂ψ_∂x1,_∂^∂ψ_x2,_∂^∂ψ_x3). Note that p0(x, ξ)is a polynomial with respect toξ. This equation can be solved with Imψ(t,x, η)≥0 and the phaseψ(t,x, η)is quasi- homogeneous of degree 1. Moreover,

ψ(t,x, η)= x, ηon, dx,η(ψ− x, η)=0 on, Imψ(t,x, η)

|η| t|η|

1+t|η| dist

x, η

|η|

,

2

, |η| ≥1.

Furthermore, there existsψ(∞,x, η) ∈ C^∞(× ˙R³)with a uniquely determined Taylor expansion at each point of such that for every compact setK ⊂ × ˙R³ there is a constantcK >0 such that

Imψ(∞,x, η)≥cK|η|

dist

x, η

|η|

,

2

, |η| ≥1.

Ifλ∈C(T^∗0),λ >0 is positively homogeneous of degree 1 andλ| <minλj, λj >0, where±iλj are the non-vanishing eigenvalues of the fundamental matrix of b, then the solutionψ(t,x, η)of (3.8) can be chosen so that for every compact set K ⊂× ˙R³and all indicesα,β,γ, there is a constantc_α,β,γ,K such that

∂x^α∂_η^β∂t^γ(ψ(t,x, η)−ψ(∞,x, η))≤c_α,β,γ,Ke^−λ(x,η)t onR₊×K.

We obtain the transport equations

T(t,x, η, ∂t, ∂x)α0=O(|Imψ|^N), ∀N,

T(t,x, η, ∂t, ∂x)αj+lj(t,x, η, α0, . . . , αj−1)=O(|Imψ|^N),∀N, j ∈N.

(3.9) It was proved in [10] that (3.9) can be solved. Moreover, there exist positively homogeneous functions of degree−j

αj(∞,x, η)∈C^∞(T^∗), j =0,1,2, . . . ,

such that αj(t,x, η) converges exponentially fast toαj(∞,x, η), t → ∞, for all j∈N0. Set

G= 1 (2π)³

∞ 0

e^{i(ψ(t,x,η)−y,η)}−t

iψt(t,x, η)α(t,x, η)+∂α

∂t(t,x, η)

dt dη

and

S= 1 (2π)³

e^{i(ψ(∞,x,η)−y,η)}α(∞,x, η)dη,

whereψt(t,x, η):= ^∂ψ_∂t(t,x, η). We can show thatGis a pseudodifferential operator of order−1 type(¹₂,¹₂),S is a pseudodifferential operator of order 0 type (¹₂,¹₂) satisfying

S+bG≡I, bS≡0.

Moreover, from global theory of complex Fourier integral operators, we can show thatS ≡

e^iϕ(x,y)ta(x,y,t)dt. Furthermore, by using some standard argument in functional analysis, we can show thatG≡G,S≡S.

Until further notice, we work in an open local coordinate patch⊂ X with real local coordinatesx=(x1,x2,x3). The following is well-known (see Chapter 5 in [10]) Theorem 3.4 With the notations and assumptions used in Theorem 3.3, let χ ∈ C^∞(R³)be equal to1near the origin. Put

G_ε(x,y) = ^∞

0

e^{i(ψ(t,x,η)−y,η)}

−t

iψ_t(t,x, η)a(t,x, η)+∂a

∂t(t,x, η)

χ(εη)dt dη,

whereψ(t,x, η), a(t,x, η)are as in(3.4). For u∈C₀^∞(), we can show that Gu:= lim

ε→0

G_ε(x,y)u(y)d y∈C^∞() and

G:C₀^∞()→C^∞(), u → lim

ε→0

G_ε(x,y)u(y)d y is continuous.

Moreover, G∈ L⁻1¹ 2,¹2

()with symbol _∞

0

e^{i(ψ(t,x,η)−x,η)}−t

iψt(t,x, η)a(t,x, η)+∂a

∂t(t,x, η)

dt ∈S⁻1¹ 2,¹2(T^∗).

We need the following (see Lemma 5.13 in [10] for a proof)

Lemma 3.5 With the notations and assumptions used in Theorem3.3, for every com- pact set K ⊂and allα ∈ N³₀,β ∈ N³₀, there exists a constant c_α,β,K > 0such that

∂^αx∂_η^β(e^{i(ψ(t,x,η)−x,η)}tψt(t,x, η))

≤c_α,β,K(1+ |η|)^|α|−|β|2 e^−tμ|η|e⁻Imψ(t,x,η)(1+Imψ(t,x, η))^1+|α|+|β|2 , (3.10) where x∈ K , t∈R₊,|η| ≥1andμ >0is a constant independent ofα,β and K .

In this work, we need

Theorem 3.6 Let L∈C^∞(X,T^1,0X⊕T^0,1X). Then, L◦G∈ L⁻

1 2 1 2,¹2(X).

Proof We work on an open local coordinate patch ⊂ X with real local coordi- natesx =(x1,x2,x3). Letl(x, η)∈ C^∞(T^∗)be the symbol of L. Then,l(x, λη)

=λl(x, η),λ >0. It is well-known (see Chapter 5 in [10]) that (L G)(x,y)≡

e^ix−y,ηα(x, η)dη,

where

α(x, η)=α0(x, η)+α1(x, η)∈S⁰1

2,¹₂(T^∗), α0(x, η)=

e^{i(ψ(t,x,η)−x,η)}(−1)l(x, ψx(t,x, η))tψt(t,x, η)a(t,x, η)dt, α1(x, η) ∈ S⁻1¹

2,¹₂(T^∗).

Herea(t,x, η) ∈ S_μ⁰(R+×T^∗),μ > 0. We only need to prove thatα0(x, η) ∈ S⁻

1 2 1

2,¹2(T^∗). Fixα, β ∈ N³₀. From (3.10), (3.5) and notice thatl(x, ψx(t,x, η))=0 at, we can check that

∂_x^α∂_η^βα0(x, η)

≤

|α|+|α|=|α|,|β|+|β|=|β|

∂x^α∂_η^β

e^{i(ψ(t,x,η)−x,η)}tψ_t(t,x, η)

×∂_x^α∂_η^β

l(x, ψ_x(t,x, η))a(t,x, η)dt

≤C_α,β

|α|+|α|=|α|,|β|+|β|=|β|

(1+ |η|)|^α|⁻|^β|

2 e^−tμ|η|e⁻¹²Imψ(t,x,η)

×(1+ |η|)¹⁻|^β| dist

x, η

|η|

,

max{0,1−|^β|^} dt

≤C_α,β

|α|+|α|=|α|,|β|+|β|=|β|

(1+ |η|)|^α|⁻|^β|

2 e^−c

t|η|2 1+t|η|

dist^x,_|η|^η ,2

×e^−tμ|η|(1+ |η|)¹⁻|β| dist

x, η

|η|

,

max{0,1−|^β|^}

dt, (3.11)

wherec>0,μ >0,C_α,β>0 andC_α,β>0 are constants.

Whenβ=0, we have

(1+ |η|)|^α|⁻|^β|

2 e^−c

t|η|2 1+t|η|

dist^x,_|η|^η ,2

×e^−tμ|η|(1+ |η|)¹⁻|β| dist

x, η

|η|

,

dt

≤c

(1+ |η|)|^α|⁻|^β|

2 1

√t(1+ |η|)⁻|^β|e^−12tμ|η|dt

≤c1

¹

1+|η|

(1+ |η|)|^α|⁻|^β|

2 1

√ (1+ |η|)⁻|^β|e^−12tμ|η|dt